Embedded Files
Materials
Materials
Volumes worksheet p. 1 (one per person)
Geometric solids of cylinder, cone, hemisphere, and sphere with equal radii (one set of solids per group)
Self-sticking sand to fill models (one per group)
Bin to contain sand (one per group)
Cylinder and Cone
Cylinder and Cone
The important dimensions for finding the volume of a cylinder and a cone are the radius of the base and the height. Our models have the same radius and height for each solid. It should be obvious that the cone is smaller than the cylinder so we might conjecture that the cylinder could fill the cone two to four times.
The sand filling the cylinder should fill the cone approximately three times.
Note: Depending on how well and consistently students pack their sand, there might be too much (or too little) sand to fill the cone exactly three times. The thickness of the models can also contribute to unevenness in filling the solids. Attaching the base can mitigate overfilling but it can be hard to fill the object completely through the small hole. When there are time constraints that prevent very careful packing, it might be helpful for students to understand that their experimentation will only provide rough estimates.
We can express the relationship between the volumes symbolically:
where the bases of the solids are congruent and the heights are equal.
By using the formula for the volume of a cylinder (product of the area of the base and height), we can express the volume of the cone in terms of its radius R and height h,
Cone and Sphere
Cone and Sphere
The important dimensions for finding the volume of a cone is the radius of the base and the height while the important dimension for the sphere is the radius. It is worth noting that the height of the cone in our models is also twice the radius of the base and the radii of both solids are equal. We should be able to conjecture that the cone is smaller than the sphere because of the way the cone tapers down to a point; it would be reasonable to say that the sphere could fill the cone one to two times.
By transferring the sand from the sphere into the cone, we find that we are able to fill the cone two times. An additional observation about the dimensions of our solids will help us derive the formula for the volume of a sphere. Since the cone has the same height as the sphere, we can express the hight h as 2R. Therefore,
Cylinder and Hemisphere
Cylinder and Hemisphere
The important dimensions of the cylinder are the radius of the base and height while the hemisphere depends only on the radius. Clearly the hemisphere is smaller than the cylinder and since the height of the cylinder is also twice the radii, we expect the cylinder to fill the hemisphere more than two times.
As we can see, the cylinder filled the hemisphere three times. Since a hemisphere of radius R is half the volume of a sphere of radius R, we can also derive the formula from this relationship:
Discussion
Discussion
Because our conclusions are based on observations of physical volume properties, finding the equation for a sphere using deduction is more challenging. The second part of the activity demonstrates how we can find the volume formula for a sphere without using calculus.
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